SOA Exam P Sample Question #597. Official problem wording is withheld; follow the source link to read it.

SOA Exam P Sample Question #597 — Verified Solution

Independent solution

How to solve this Uniform Distribution question

Answer in brief

The no-deductible mean identifies the uniform support as [1,19]. A deductible of 4 leaves a positive excess over a 15-unit interval; integrating that excess against density 1/18 gives 225/36=6.25, so choice B is correct.

Setup

Use the no-deductible expected payment to recover the lower endpoint of the uniform loss distribution.

\frac{a+19}{2}=10

a=1

Model

The loss therefore has constant density 1/18 on [1,19], and the second policy pays only the positive excess above 4.

X\sim\operatorname{Uniform}(1,19)

Y=(X-4)_+

Compute

Integrate the payment over the part of the support above the deductible.

E[Y]=\int_{4}^{19}(x-4)\frac{1}{18}\,dx

E[Y]=\frac{(19-4)^2}{36}=6.25

Answer

The expected payment under the deductible policy is 6.25.

\boxed{6.25\quad\text{(B)}}

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis subtracts the deductible directly from the original mean, 10-4=6, and allows negative payments for losses below 4.
CThis is the mean excess conditional on a payment, (19-4)/2=7.5, and omits the probability that a payment occurs.
DThis scales the original mean by the fraction (19-4)/19, treating the support as if it began at zero instead of integrating the excess.
EThis multiplies the original mean 10 by P(X>4)=15/18 but does not subtract the deductible from paid losses.

Original practice · fully worked

Original variant: research-vessel repair threshold

A research vessel's repair-cost index X is uniformly distributed from 0 to 24. A reserve pays the amount of X above an unknown activation threshold d and pays zero otherwise. The expected reserve payment is 3. Determine d.

A
3
B
9
C
12
D
18
E
21

Variant answer in brief

For a uniform index on [0,24], the expected positive excess over d is (24-d)^2/48. Equating this to 3 gives 24-d=12 and hence d=12, so choice C is correct.

Setup

Represent the reserve payment as the positive part above the unknown threshold.

X\sim\operatorname{Uniform}(0,24)

R=(X-d)_+

Model

Integrate the excess payment using the constant uniform density.

E[R]=\int_d^{24}(x-d)\frac1{24}\,dx

E[R]=\frac{(24-d)^2}{48}

Compute

Set the expected payment equal to its specified value and retain the threshold within the support.

\frac{(24-d)^2}{48}=3

(24-d)^2=144

d=12

Answer

The reserve activates at a cost index of 12.

\boxed{d=12\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #597 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.